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17 votes
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Comparing method of differentiation in variational quantum circuit

Both finite differences and the parameter-shift rule can be used to compute quantum gradients on quantum hardware. However, there are several reasons that lead to the parameter-shift rule being ...
Josh Izaac's user avatar
12 votes
Accepted

What's the role of mixer in QAOA?

Probably the easiest way to understand this is to pretend that the mixer is NOT there and see what happens. So, let's assume you have some initial state $\lvert \psi \rangle = \sum_x \psi_x \lvert x \...
Gianni Mossi's user avatar
9 votes
Accepted

Is there a general method of expressing optimization problem as a Hamiltonian?

As requested in the comments, here is a worked example. The main body deals with minimizing $f(x)$ for a specific problem. At the bottom follows a brief discussion of constraints then a brief ...
bRost03's user avatar
  • 579
8 votes
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How to maximise over linear functionals of quantum channels?

For the specific linear function you are interested in, the solution turns out to be trivial: you can take the channel to be $N_{X\rightarrow Y}(\rho) = \operatorname{Tr}(\rho) |\psi\rangle\langle \...
John Watrous's user avatar
  • 6,147
8 votes
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What are the differences between the different transpiler optimization levels in qiskit

These levels are provided via "preset passmanagers" in Qiskit. These are simple-to-use transpiler pipelines, but you can also build your own passmanager pipeline. You can see what each does by ...
Ali Javadi's user avatar
  • 1,632
7 votes
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What is the difference between QAOA and Quantum Annealing?

One of the advantages, as stated in the paper you linked, is that with QAOA you can increase the precision arbitrarily, whereas QA will only find the solution with probability 1 as $T \to \infty$ ...
Andrew O's user avatar
  • 1,749
7 votes

What exactly happening in QAOA in a general way?

A precursor to the canonical QAOA is the Quantum Adiabatic Algorithm (QAA). Since we want to end up in the ground state of the Cost Hamiltonian ($H_C$) but don't know how to construct it, we exploit ...
Faiyaz Hasan's user avatar
7 votes
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How to show mathematically the equivalency between Ising Model and QUBO?

The relation between "Ising" and binary variables is following $$ x_i = \frac{1 + s_i}{2}, $$ where $s_i$ is a spin and $x_i$ is a binary variable. Clearly setting $s_i = -1$ leads to $x_i = ...
Martin Vesely's user avatar
7 votes
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T-depth in Qiskit

The depth method can be customized with a gate subset you want to consider. To compute the $T$-depth you most likely want to include both $T$ and $T^\dagger$ gates. ...
Nikita Nemkov's user avatar
6 votes
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QAOA for MaxCut - Algorithm motivation

What motivated this construction is mentioned in the original paper (section VI): adiabatic quantum computing. This construction is basically a Trotterized version of the evolution by the time ...
cnada's user avatar
  • 4,764
6 votes
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How to convert QUBO problem to Ising Hamiltonian?

Maybe this will help. Let's take a simple case: $$f(x_1, x_2) = -2x_1 x_2$$ Then it is minimum when $x_1 = x_2 = 1$. Now let's take this Hamiltonian: $$H_f = -2Z \otimes Z$$ The Hamiltonian is minimum ...
Davit Khachatryan's user avatar
6 votes

Complexity of $n$-Toffoli with phase difference

(This answer uses ancillae and feedback) Does anyone know if there has been any improvement on the decomposition of the general n-qubits control X with phase differences in terms of elementary gates ...
Craig Gidney's user avatar
  • 39.2k
6 votes
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Under what conditions the minimum eigengap is non-zero?

I think this is formally undecidable. In detail, Cubitt, Perez-Garcia, and Wolf (arxiv, Nature) reduced the problem of determining the gap of a translationally-invariant Hamiltonian to the problem of ...
Mark Spinelli's user avatar
6 votes

Does QAO Ansatz have any better performance guarantees than QAOA?

The statement that "as $p \rightarrow \infty$, the minimum of the objective function is reached" is not correct. In fact, it is a pretty meaningless statement. Commonly, a QAOA circuit has $...
MonteNero's user avatar
  • 2,704
5 votes
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How to tell if the ground states of two Hamiltonians are solutions of the same optimization problem?

Short Answer: It is potentially hard (as bRost03 indicates in the comments). To be precise, coNP-hard. Longer Answer: In adiabatic quantum computation, the ground-...
Niel de Beaudrap's user avatar
5 votes

Does the Qiskit ADMM optimizer really run on quantum computers?

The ADMM optimizer is a classical optimizer that will be execute on the classical computer. Nowadays, because of the limitation of the hardware, we see a lot of hybrid quantum-classical algorithms. In ...
KAJ226's user avatar
  • 13.9k
5 votes
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maximization of trace between two operators with respect to different norm constraints

If I understand your question correctly, you're trying to prove something that is false. Consider the operator $$ Y = \begin{pmatrix} 2 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 &...
John Watrous's user avatar
  • 6,147
5 votes
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Cost of SWAP gate

Swaps are never truly free. The papers you linked are just ignoring the cost of routing, and then hiding the cost of swapping in that ignorance. We ignore all concerns of layout and communication ...
Craig Gidney's user avatar
  • 39.2k
5 votes
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Choosing a different optimizer when running QAOA in qiskit

The optimizers in Qiskit need to be instantiated then you can call their minimize() method. ...
Steve Wood's user avatar
  • 1,508
5 votes
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Does the gradient commute with the partial trace?

In brief: it is because differentiation and (partial) traces are linear operations, so we can pull them through each other. Physically, one might interpret this as the fact that it doesn't matter ...
Banach space fan's user avatar
5 votes
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Optimization in QAOA should only return global minimum?

Since QAOA is an approximation method I would assume it is impossible to find the global minimum each time but others seem to disagree... There's a difference between being guaranteed to find the ...
Tristan Nemoz's user avatar
  • 6,897
4 votes
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Barren plateaus in quantum neural network training landscapes

First: The paper references [37] for Levy's Lemma, but you will find no mention of "Levy's Lemma" in [37]. You will find it called "Levy's Inequality", which is called Levy's Lemma in this, which is ...
user1271772 No more free time's user avatar
4 votes
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Devising "structured initial guesses" for random parametrized quantum circuits to avoid getting stuck in a flat plateau

I am not an expert but I read a few papers and here is what I have found. Similarly to NN, people found strategies to avoid this issue with the gradients. Basically, for some problems, you can use ...
Enrico's user avatar
  • 473
4 votes
Accepted

Minimum number of CNOTs for Toffoli with non-adjacent controls

Here is the best construction I've found. It uses 8 CNOTs. I verified this circuit in Quirk using the channel-state duality and a known-good inverse. The target is the middle qubit. None of the ...
Craig Gidney's user avatar
  • 39.2k
4 votes
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Resources on hybrid quantum-classical algorithms applied to combinatorial optimization problems

So for hybrid quantum-classical algorithms, I suggest looking at : The Quantum Approximate Optimization Algorithm Variational hybrid quantum-classical algorithms that include the so famous ...
cnada's user avatar
  • 4,764
4 votes
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Minimum number of CNOTs for a 4-qubit increment on a planar grid

Here is the best circuit I've found. It uses 14 CNOTs. Note that this circuit is not using a linear layout! It is placed on the grid like this: ...
Craig Gidney's user avatar
  • 39.2k
4 votes

How to explain that I get a value lower than the smallest possible through minimization procedure in VQE?

The proof of the variational theorem (the theorem that the ground state energy is the lowest possible energy you can get from $\frac{\langle \psi|H|\psi\rangle}{ \langle \psi | \psi \rangle}$) is ...
Randy Hyatt's user avatar
4 votes
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How does the classical optimization of the angles $\gamma$ and $\beta$ in QAOA work?

$\langle \psi_p(\gamma,\beta)|H|\psi_p(\gamma,\beta)\rangle$ is basically the function evaluation step during the optimization. If you use a gradient-free optimizer, then it uses this information to ...
cnada's user avatar
  • 4,764
4 votes
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Can QAOA be considered as simulation of a quantum annealer on a gate-based quantum computer?

If you use infinite depth then QAOA can be consider as quantum annealer on gate-based. The authors of QAOA original paper probably deduce it from quantum annealing. What I mean by infinite depth is ...
KAJ226's user avatar
  • 13.9k
4 votes

Why QAOA with $p \rightarrow \infty $ gives the optimal solution?

The Quantum Approximate Optimization Algorithm is closely related to the Quantum Adiabatic Algorithm. Let's say we have a simple Hamiltonian (in our case $H_B$) with a known ground state and another ...
GiannisKol's user avatar

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